-4, -1, 2, 5, ..., find the 15th term Step by Step

Step by step

The question

Question: −4, −1, 2, 5, …: find term 15.

A sequence is a list of numbers in order.

−4, −1, 2, 5, ...

1 Find the difference

We add 3 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 3.

d = 3

2 Find the rule

Term 1 is −4.

a₁ = −4

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

Term 2: we add 3 one time.

a₂ = −4 + 1 × 3

Term 3: we add it two times.

a₃ = −4 + 2 × 3

We add 3 one time less than the term number.

times we add 3 = term number − 1

So for term n, we add 3, (n − 1) times.

a_n = −4 + (n − 1) × 3

This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.

a_n = a₁ + (n − 1) × d

3 Find term 15

For term 15, n is 15.

We put 15 in place of n.

a₁₅ = −4 + (15 − 1) × 3

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

15 minus 1 is 14.

= −4 + 14 × 3

Times comes before plus.

14 times 3 is 42.

= −4 + 42

Start at −4.

Go up 42.

We land on 38.

a₁₅ = 38

So term 15 is 38.

4 Check

Let's check the rule with a term we know.

Put n = 4.

a₄ = −4 + (4 − 1) × 3 = −4 + 3 × 3 = −4 + 9

= 5

We get 5.

That is term 4 in the list.

So the rule is right!

Let's look back.

First: find the difference.

Then: find the rule.

Then: find term 15.

Last: check.

Answer